Laplace Transform Method Research Articles

Método de perturbación con transformada de Laplace para resolver problemas no lineales de múltiples soluciones, con condiciones a la frontera mixtas y Neumann

El campo de las ecuaciones diferenciales ha cobrado auge en la actualidad por el desarrollo científico y tecnológico. Por esta situación, el estudio de nuevas metodologías para solucionarlas se ha vuelto importante. A partir de la combinación del método de Laplace Transform (LT) y el método de perturbación (PM) este trabajo presenta el método LT-PM, y su motivación se encuentra en la aplicación conocida de la LT a ecuaciones diferenciales ordinarias lineales. El objetivo de este trabajo fue presentar una modificación del método de perturbación (PM), el método de perturbación con transformada de Laplace (LT-PM), con el fin de resolver problemas perturbativos no lineales, con condiciones a la frontera definidas en intervalos finitos. La metodología consistió en aplicar LT a la ecuación diferencial por resolver y después de asumir que la solución de la misma se puede expresar como una serie de potencias de un parámetro perturbativo, se obtiene la solución del problema aplicando sistemáticamente la transformada inversa de Laplace. Los principales resultados de este trabajo se muestran a partir de dos casos de estudio presentados, donde se observa que LT-PM es potencialmente útil para encontrar soluciones múltiples de problemas no lineales. Además, LT-PM mejora la aplicabilidad del método de perturbación en algunos casos de condiciones a la frontera mixtas y de Neumann, donde PM simplemente no funciona. Con el fin de verificar la exactitud de los resultados obtenidos, se calculó su error residual cuadrático (SRE), el cual resultó muy bajo, de donde se dedujo su precisión y la potencialidad de LT-PM. Se concluye que si bien el método propuesto resulta eficiente en los casos particulares presentados, se espera que sea una herramienta potencialmente eficiente y útil para otros casos de estudio, particularmente, en aquellos relacionados con aplicaciones prácticas en ciencias e ingeniería.

English

English

Review of wound roll stress in roll-to-roll manufacturing of flexible electronics

Flexible electronics has been widely used in many main-stream applications like flexible display, thin-film solar cell, RFID tags, flexible sensors, and so on. Roll-to-roll (R2R) is a kind of method by which flexible substrate can be processed in a continuous way. This kind of method saves time, cost, reduces delay time and increases efficiency, throughput, performance, etc. The typical R2R system mainly consists of unwinding unit, feeding unit and winding unit, where deviation rectification control, tension control, vibration control and position control are needed. By taking full advantages of the deformation characteristics of flexible electronics, the R2R process has become one of the most effective ways in manufacturing flexible electronics. The winding unit is an essential part in the R2R system and the control of the wound roll stress in winding process is one of the main factors affecting the quality and productivity of flexible electronics. A tiny wound roll stress will cause difficulties in storage and transportation. The substrates or the electronic circuits mounted on it may be broken when the wound roll stress is too large. This paper begins with the discussion of common issues and the influence of wound roll stress in winding process. Currently, the web tension and its lateral position are two factors that mainly affect the wound roll stress. The wound roll stress will increase with the augment of the web tension. While a variation of lateral position can cause non-uniform wound roll stress in cross machine direction. Subsequently, the modelling methods and solution methods based on elastic mechanics, Laplace transformation, discrete modelling and finite element method are generalized. The four models are called elastic winding model, viscoelastic winding model, pseudo winding model and FEM winding model respectively. The elastic model is derived to predict wound roll stress of the elastic web. The viscoelastic model is developed to study how the wound roll stress changes with time after winding. In the pseudo model and the FEM model, the effect of thickness variation along the width direction is considered. Then this review analyses the effects of winding speed-induced inertia force, web deformation and its inhomogeneous thickness, temperature, air entrainment, structure of the nip roller and the core shaft on the wound roll stress. The taper tension control and constant tension control methods for wound roll stress and engineering applications in RFID tag packaging, seal lamination and flexible OLED manufacturing are also presented. Finally, this paper forecasts the key problems need to be further studied for the wound roll stress in winding process. The study of wound roll stress may focusing on the following aspects: (1) Conducting wound roll stress studies that considering the effect of thickness variation in machine direction; (2) usually we consider the wound roll as an axisymmetric cylinder, so it is necessary to develop a winding model to deal with the helical structure of the wound roll; (3) when deriving winding models, its necessary to consider the discontinuous contact between layers.

Application of dispersant to slow down physical hardening process in asphalt binder

Physical hardening in asphalt binder is recognized as one of the main factors leading to premature and excessive cracking in cold regions and high-altitude areas. To slow down the physical hardening process in asphalt binder, octyl aniline (OA), which is a commonly available non-ionic dispersant, was added to the asphalt binder. The effectiveness of OA in retarding the physical hardening process of the base asphalt binder was evaluated with an extended bending beam rheometer and critical-cracking temperature tests. Further, a thermal analysis was conducted to determine the glass transition temperature. The results showed that the one-step Laplace transformation method is an alternative to the asphalt binder thermal-stress calculation. Moreover, adding a sufficient amount of OA significantly decreased the magnitude of physical hardening in the base asphalt binder. The amphiphilic structure of OA can inhibit the aggregation of asphaltenes to a certain extent, which might be the main reason leading to a lower magnitude of physical hardening in the OA-modified asphalt binder.

STUDY OF HALL AND SORET EFFECT ON MHD FLOW WITH A RAMPED PLATE TEMPERATURE OF AN EXPONENTIALLY ACCELERATED VERTICAL PLATE EMBEDDED IN A POROUS MEDIUM

The present literature deals with the study of MHD flow of an incompressible viscous electrically conducting fluid along an exponentially accelerated infinite vertical plate in a rotating system. The effects of thermodiffusion, thermal radiation and Hall current, are analysed with ramped and isothermal plate conditions. The non-dimensional coupled partial differential equations of the model are solved analytically by using Laplace Transform method with the help of Heaviside step function; thus the expression for velocity field, temperature field, concentration field, skin friction coefficient, Nusselt number and Sherwood number are obtained. The influence of physical parameters such as m (Hall parameter), Sr (Soret number), K (Darcy permeability), λ (exponential parameter), Ω (rotation parameter) and R (radiation parameter) on these fields are discussed in detail through graphs.

Solutions of the Singular IVPs of Lane-Emden type equations by combining Laplace transformation and perturbation technique

AbstractIn this paper, we propose an efficient method to solve linear and nonlinear singular initial value problems of Lane-Emden type equations by combining Laplace transformation and homotopy perturbation methods. The method is based upon Laplace transform, polynomial series and perturbation technique. Several examples, including some well-known Lane-Emden problems, are presented to show the ability and accuracy of the modify method.

Solutions of Stokes Second Problem for Casson Fluid over an Infinite Plate: A Comparison of the Laplace and Fourier Transforms

The unsteady flow of Casson fluid over a plane wall, which is initially at rest and the plate suddenly starts oscillations in its own plane is studied by mean of Laplace transform method and Fourier transform method. The solutions are obtained as a sum of steady and transient solutions, which satisfy the governing equation and all impose initial and boundary conditions. The comparison of the solutions obtained by two integral transform methods is presented in graphical illustrations. The transient parts are found in terms of definite integrands whose integral are oscillatory function. Finally, these solutions are graphically shown and discussed.

Study on longitudinal vibration of a pile with variable sectional acoustic impedance by integral transformation

A study on longitudinal vibration of a pile with variable sectional acoustic impedance subjected to arbitrary form of external excitation is conducted. The analytical solution of transfer function is primarily solved by Laplace transform, and then its corresponding impulse response function is solved by residue method of inverse Laplace transformation. With impulse response function, the analytical solution of response at pile top can be obtained by convolution calculation, which overcomes the limit of previous analytical solutions subjected to prescribed time-harmonic load. The accuracy of presented solution is verified by compared with existing solutions subjected to semi-sine excitation and low-strain test of model piles. To take advantage of presented solution, this paper also proposes a case study of imposed excitation considering superimposed high-frequency interference component, and some conclusions derived from this case can provide practical guidance for engineering implementation.

An improved maximum entropy method via fractional moments with Laplace transform for reliability analysis

The fractional moment-based maximum entropy method (FM-MEM) attracts more and more attention in reliability analysis recently, comparing with the common integer moment-based maximum entropy method. In the present paper, a new approach for reliability analysis is proposed from the improvement of the fractional moment-based maximum entropy method via the Laplace transformation and dimension reduction method (DRM). Different with the traditional FM-MEM with a double-loop multivariate optimization formulation, the proposed method introduces a single-loop univariate optimization algorithm through solving a set of linear equations. Firstly, the single-loop algorithm is proposed for bounded positive random variable and then is extended to arbitrary positive random variable with the help of Laplace transform. Then, a univariate strategy is given to simplify the calculation for the unknown parameters in the single-loop algorithm. Through the improvement, not only the proposed method can predict failure probability accurately but also the computational cost is considerably reduced. Due to the univariate optimization strategy, the proposed method is more robust than common reliability analysis methods, which are sensitive to the initial points, such as the traditional FM-MEM and the first-order reliability method. Several numerical examples are studied to illustrate the efficiency, robustness, and accuracy of the proposed method for the prediction of the failure probability in comparison with other methods.

Dynamic Response of Multi-cracked Beams Resting on Elastic Foundation

Cracks cause to change dynamic response of beams and make discontinuity in slope of the deflection of the beams. The dynamic analysis of the Euler-Bernoulli cracked beam on the elastic foundation subjected to the concentrated load is presented in this paper. The stiffness of the elastic foundation and elastic supports influence on vibrational characteristics of the cracked beam. The Dynamic Green Function is applied to solve the governing equation. Thus, the dynamic response of the cracked beam is determined by Laplace Transform method. The effects of depth and location of the crack on natural frequency and deflection of the cracked beam on an elastic foundation are evaluated. In order to demonstrate the capability of the present approach, several numerical examples are worked out and the results are discussed.

Transient responses of a hollow cylinder under thermal and chemical shock based on generalized diffusion-thermoelasticity with memory-dependent derivative

The accurate description of the interactions of temperature, concentration and strain fields is of great importance for the fabrication of micro/nano-electromechanical devices. This article aims to conduct an analytical study of transient responses of a hollow cylinder under thermal and chemical shock in the context of generalized diffusion–thermoelasticity with memory-dependent derivative. The outer surface of the cylinder is assumed to be traction free and subjected to transient thermal and chemical shock loadings, while its inner surface is taken to be in contact with a rigid surface and is thermally isolated. The governing equations are obtained and solved by using Laplace transformation methods. The dimensionless results of temperature, chemical potential, displacement, stress, concentration, heat flux, and diffusion flux along the radial direction of the cylinder are obtained and illustrated graphically. Parametric studies are also performed to evaluate the influences of time delay and kernel function on transient thermoelasto-diffusive responses.

Prediction of 3D grinding temperature field based on meshless method considering infinite element

A three-dimensional numerical model to calculate the grinding temperature field distribution is presented. The finite block method, which is developed from meshless method, is used to deal with the stationary and the transient heat conduction problems in this paper. The influences of workpiece feed velocity, cooling coefficient, and the depth of cut on temperature distribution are considered. The model with temperature-dependent thermal conductivity and specific heat is presented. The Lagrange partial differential matrix from the heat transfer governing equation is obtained by using Lagrange series and mapping technique. The grinding wheel-workpiece contact area is assumed as a moving distributed square heat source. The Laplace transformation method and Durbin’s inverse technique are employed in the transient heat conduction analysis. The results of the developed model are compared with others’ finite element method solutions and analytical solutions where a good agreement is demonstrated. And the finite block method was proved a better convergence and accuracy than finite element method by comparing the ABAQUS results. In addition, the three-dimensional infinite element is introduced to perform the thermal analysis, and there is a great of advantages in the simulation of large boundary problems.

THE FRACTIONAL LAPLACE TRANSFORM SOLUTION FOR FRACTIONAL DIFFERENTIAL EQUATION THE OSCILLATOR IN THE PRESENCE OF AN EXTERNAL FORCES

The Fractional Laplace transform is applied to derive the solution of fractional differential equation of order governing the motion of the Harmonic Oscillator in a Non-Resisting Medium. The Fractional Laplacetransform method is to provide a potential tool to solve such a type of Fractional differential equation. Further consider oscillator in the presence of an external driving force of is in the fractional form and investigate some interesting features of the given equation. Also discuss the Numerical form of the oscillator in a r. The Fractional Laplace transform is applied to derive the solution of fractional differential equation of order governing the motion of the Harmonic Oscillator in a Non-Resisting Medium. The Fractional Laplacetransform method is to provide a potential tool to solve such a type of Fractional differential equation. Further consider oscillator in the presence of an external driving force of is in the fractional form and investigate some interesting features of the given equation. Also discuss the Numerical form of the oscillator in a resisting medium and discuss some cases of oscillation fractional differential equations for external driving force. esisting medium and discuss some cases of oscillation fractional differential equations for external driving force.

Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries

Laplace transform method (LTM) has a lot of applications in the evaluation of European-style options and exotic options without early exercise features. However the Laplace transform methods for pricing American options have unsatisfactory accuracy and suffer from the instability. The aim of this paper is to develop a Laplace transform method for pricing American Strangles options with the underlying asset price following the constant elasticity volatility (CEV) models. By approximating the free boundaries, the Laplace transform is taken on a fixed space region to replace the moving boundaries space. After solving the linear system in Laplace space, Gaver-Stehfest formula (GSF) and hyperbola contour integral method (HCIM) are applied to compute the Laplace inversion. Numerical results show that the LTM-HCIM outperform the LTM-GSF in regard to the accuracy and stability for the option values.

Thermal Radiation on Transient Laminar Gray Gas Flow Past an Oscillating Vertical Plate with Variable Temperature

Authors have presented an analytical approach on transient laminar gray gas flow past an oscillating vertical flat plate in the presence of natural convection and radiation. The fluid is considered to be a gray, absorbing-emitting radiation but non- scattering medium. The Rosseland flux approximation is employed to simulate radiation heat transfer contribution. We have also studied Stoke’s first and second problem to justify the physical significance on this problem. This problem is solved by employing Laplace Transform method. Numerical results of velocity and temperature distributions are depicted graphically. Also, numerical results of frictional shearing stress and critical Grashof number are presented in tables.

Heavy-Light Mesons in the Nonrelativistic Quark Model Using Laplace Transformation Method

An analytic solution of theN-dimensional radial Schrödinger equation with the combination of vector and scalar potentials via the Laplace transformation method (LTM) is derived. The current potential is extended to encompass the spin hyperfine, spin-orbit, and tensor interactions. The energy eigenvalues and the corresponding eigenfunctions have been obtained in theN-dimensional space. The present results are employed to study the different properties of the heavy-light mesons (HLM). The masses of the scalar, vector, pseudoscalar, and pseudovector forB,Bs,D, andDsmesons have been calculated in the three-dimensional space. The effect of the dimensional number space is discussed on the masses of the HLM. We observed that the meson mass increases with increasing dimensional space. The decay constants of the pseudoscalar and vector mesons have been computed. In addition, the leptonic decay widths and branching ratio for theB+,D+, andBs+mesons have been studied. Therefore, the used method with the current potential gives good results which are in good agreement with experimental data and are improved in comparison with recent theoretical studies.

Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order

AbstractSystematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.

Evaluation of Non-Fourier Heat Transfer on Temperature Evolution in an Aquifer Thermal Energy Storage System

The heat transport in fractured porous rocks is classically modeled employing the advection–dispersion equation (ADE). However, the nature of heat transfer in fractured reservoir rocks may not be represented by the effective medium properties when the ADE formulation is adopted. In this study, a modified mathematical model describing non-Fourier heat transport in an aquifer thermal energy storage (ATES) system is proposed employing the fractional calculus theory. This mathematical model incorporates the effect of heat losses to the surrounding impermeable rock formations. The Laplace transformation method is applied to derive the semi-analytical solutions describing the dimensionless temperature evolution in the confined aquifer, and the surrounding impermeable rocks (i.e., underlying and overlying rocks). Detailed parametric studies are performed to investigate the role of the introduced parameters, i.e., the fractional order of differentiation, generalized friction coefficient and aquifer pseudo-effective thermal conductivity on the propagation of heat within the ATES system. Computations performed on the derived solutions demonstrate that the temperature profiles in the confined aquifer and the surrounding rocks are influenced by the magnitude of the respective fractional exponents. In addition, observation of the temperature profiles within the thermally perturbed zones demonstrates that larger values of the fractional order of differentiation lead to efficient heat transfer within the ATES system. Furthermore, analysis of the results indicates that the impact of the aquifer pseudo-effective thermal conductivity on the temperature propagation in the ATES system is limited to the aquifer only. The derived solutions will find widespread application in designing and simulating the heat injection performance in an ATES system and assessing the influence of non-Fourier heat transport and geological parameters on temperature transients through porous media.

Application of Numerical Inverse Laplace Transform Methods for Simulation of Distributed Systems with Fractional-Order Elements

The paper presents a computationally efficient method for modeling and simulating distributed systems with lossy transmission line (TL) including multiconductor ones, by a less conventional method. The method is devised based on 1D and 2D Laplace transforms, which facilitates the possibility of incorporating fractional-order elements and frequency-dependent parameters. This process is made possible due to the development of effective numerical inverse Laplace transforms (NILTs) of one and two variables, 1D NILT and 2D NILT. In the paper, it is shown that in high frequency operating systems, the frequency dependencies of the system ought to be included in the model. Additionally, it is shown that incorporating fractional-order elements in the modeling of the distributed parameter systems compensates for losses along the wires, provides higher degrees of flexibility for optimization and produces more accurate and authentic modelling of such systems. The simulations are performed in the Matlab environment and are effectively algorithmized.

Pressure transient analysis of multi-fractured horizontal wells in tight oil reservoirs with consideration of stress sensitivity

Multi-fractured horizontal well (MFHW) is an effective technique to develop unconventional reservoirs. Complex fracture network around the well and hydraulic fractures is formed during the fracturing process. Fracture network and hydraulic fractures are the main seepage channel which is sensitivity to the effective stress. However, most of the existing models do not take the effect of stress sensitivity into account. In this study, a new analytical model was established for MFHW in tight oil reservoirs based on the trilinear flow model. Fractal porosity and permeability were employed to describe the heterogeneous distribution of the complex fracture network, and the stress sensitivity of fracture was considered in the model. The Pedrosa substitution and perturbation method were applied to eliminate the nonlinearity of the model. By using the Laplace transformation method, the analytical solutions in Laplace domain were obtained. Then, validations were performed to show that the model is valid. Finally, sensitivity analysis was discussed. The presented model provides a new approach to the estimation of fracturing effect and can be also utilized for recognizing formation properties of tight oil reservoirs.